Table Of ContentMathematics in Industry 20
Peter Benner Editor
System Reduction
for Nanoscale
IC Design
MATHEMATICS IN INDUSTRY 20
Editors
HansGeorgBock
FrankdeHoog
AvnerFriedman
ArvindGupta
AndréNachbin
TohruOzawa
WilliamR.Pulleyblank
TorgeirRusten
FadilSantosa
JinKeunSeo
Anna-KarinTornberg
Moreinformationaboutthisseriesathttp://www.springer.com/series/4650
Peter Benner
Editor
System Reduction for
Nanoscale IC Design
123
Editor
PeterBenner
MaxPlanckInstituteforDynamics
ofComplexTechnicalSystems
Magdeburg,Germany
ISSN1612-3956 ISSN2198-3283 (electronic)
MathematicsinIndustry
ISBN978-3-319-07235-7 ISBN978-3-319-07236-4 (eBook)
DOI10.1007/978-3-319-07236-4
LibraryofCongressControlNumber:2017942739
MathematicsSubjectClassification(2010):94-02,65L80,94C05
©SpringerInternationalPublishingAG2017
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof
thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,
broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation
storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology
nowknownorhereafterdeveloped.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook
arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor
theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany
errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional
claimsinpublishedmapsandinstitutionalaffiliations.
Printedonacid-freepaper
ThisSpringerimprintispublishedbySpringerNature
TheregisteredcompanyisSpringerInternationalPublishingAG
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
Preface
The ongoing miniaturization of devices like transistors used in integrated circuits
(ICs) has led to feature sizes on the nanoscale. The Intel Core 2 (Yorkfield), first
presentedin2007,wasproducedusing45nmtechnology.Recently,productionhas
reached 14nm processes, e.g., in the Intel Broadwell, Skylake, and Kaby Lake
microprocessors. Although the main principles in IC design and production are
thoseofmicroelectronics,nowadays,onethereforespeaksofnanoelectronics.
Withminiaturizationnowreachingdouble-digitnanometerlengthscalesandthe
hugenumberofsemiconductordevicesemployed,whichresultinacorrespondingly
significant rise in integration density, the influence of the wiring and supply
networks (interconnect and power grids) on the physical behavior of an IC can
no longer be neglected and must be modeled with the help of dedicated network
equationsinthecaseofcomputersimulations.Furthermore,criticalsemiconductor
devices can often no longer be modeled by substitute schematics as done in the
past, using, e.g., the Partial Element Equivalent Circuit (PEEC) method. Instead,
complexmathematicalmodelsareused,e.g.,thedrift-diffusionmodel.Inaddition
to shortened production cycles, these developments in the design of new nano-
electronicICsnowincreasinglyposechallengesincomputersimulationsregarding
the optimizationand verificationof layouts.Evenin the developmentstage, it has
become indispensable to test all crucial circuit properties numerically. Thus, the
fieldofcomputationalnanoelectronicshasemerged.
The complexity of the mathematical models investigated in computational
nanoelectronicsisenormous:smallpartsofanICdesignalonemayrequiremillions
of linear and nonlinear differential-algebraic equations for accurate modeling,
allowingthepredictionofitsbehaviorinpractice.Thus,thefullsimulationofanIC
designrequirestremendouscomputationalresources,whichareoftenunavailableto
microprocessordesigners.Inshort,onecouldjustifiablyclaimthattheperformance
of today’s computers is too low to simulate their successors!—a statement that
has been true for the last few decades and is debatably still valid today. Thus,
thedimensionreductionofthemathematicalsystemsinvolvedhasbecomecrucial
over the past two decades and is one of the key technologies in computational
nanoelectronics.
v
vi Preface
The dimension or model reductionat the system level, or system reduction for
short, is mostly done by mathematicalalgorithms,which producea much smaller
(oftenbyfactorsof100upto10,000)modelthatreproducesthesystem’sresponse
to a signal up to a prescribed level of accuracy. The topic of system reduction
in computational nanoelectronics is the focus of this book. The articles gathered
herearebasedonthefinalreportsforthenetworkSystemReductioninNanoscale
IC Design (SyreNe), supported by Germany’s Federal Ministry of Education and
Research (BMBF) as part of its Mathematics for Innovations in Industry and
Services program. It was funded between July 1, 2007, and December 31, 2010
(see syrene.org for a detailed description) and continued under the name Model
ReductionforFastSimulationofnewSemiconductorStructuresforNanotechnology
and Microsystems Technology (MoreSim4Nano) within the same BMBF funding
schemefromOctober1,2010,untilMarch31,2014(seemoresim4nano.org).
The goal of both research networks was to develop and compare methods for
system reductionin the designof high-dimensionalnanoelectronicICs and to test
the resulting mathematical algorithms in the process chain of actual semiconduc-
tor development at industrial partners. Generally speaking, two complementary
approaches were pursued: the reduction of the nanoelectronic system as a whole
(subcircuitmodelcoupledtodeviceequation)bymeansofaglobalmethodandthe
creationofreducedordermodelsforindividualdevicesandlargelinearsubcircuits
which are linked to a single reduced system. New methods for nonlinear model
reduction and for the reduction of power grid models were developed to achieve
this.
The bookconsistsof five chapters,introducingnovelconceptsfor the different
aspectsofmodelreductionofcircuitanddevicemodels.Theseinclude:
(cid:129) Model reduction for device models coupled to circuit equations in Chap.1 by
Hinze,Kunkel,Matthes,andVierling
(cid:129) Structure-exploiting model reduction for linear and nonlinear differential-
algebraic equations arising in circuit simulation in Chap.2 by Stykel and
Steinbrecher
(cid:129) The reduced representation of power grid models in Chap.3 by Benner and
Schneider
(cid:129) Numeric-symbolic reduction methods for generating parameterized models of
nanoelectronicsystemsinChap.4bySchmidt,Hauser,andLang
(cid:129) Dedicated solvers for the generalized Lyapunov equations arising in balanced
truncation based model reduction methods for circuit equations in Chap.5 by
BollhöferandEppler
Theindividualchaptersdescribethenewalgorithmicdevelopmentsintherespec-
tive researchareasoverthe course ofthe project.Theycan be readindependently
ofeachotherandprovideatutorialperspectiveontherespectiveaspectsofSystem
Reductionin NanoscaleIC Design related to the sub-projectswithin SyreNe.The
aimis to comprehensivelysummarizethe latestresearchresults, mostlypublished
in dedicatedjournalarticles, andto presenta numberof new aspectsneverbefore
Preface vii
published.Thechapterscanserveasreferenceworks,butshouldalsoinspirefuture
researchincomputationalnanoelectronics.
Iwouldliketotakethisopportunitytoexpressmygratitudetotheprojectpart-
ners Matthias Bollhöfer and Heike Faßbender (both from the TU Braunschweig),
Michael Hinze (University of Hamburg), Patrick Lang (formerly the Fraunhofer-
Institut für Techno-und Wirtschaftsmathematik(ITWM), Kaiserslautern), Tatjana
Stykel(attheTUBerlinduringtheprojectandnowattheUniversityofAugsburg),
Carsten Neff (NEC EuropeLtd. back then), Carsten Hammer (formerly Qimonda
AG and thenInfineonTechnologiesAG), andPeter Rotter (InfineonTechnologies
AGbackthen).OnlytheircooperationwithinSyreNeandtheirvaluedworkinthe
variousprojectsmadethisbookpossible.
Furthermore,IwouldliketoparticularlythankAndréSchneider,whohelpedin
countless ways duringthe preparationof this book. This includesthe LATEX setup
aswellasindexingandresolvingmanyconflictsinthebibliographies.Withouthis
help, I most likely never would have finished this project. My thanks also go to
RuthAlleweltandMartinPetersofSpringer-Verlag,whowereverysupportiveand
encouraging throughout this project. Their endless patience throughout the many
delaysinthefinalphasesofpreparingthebookisgreatlyappreciated!
Magdeburg,Germany PeterBenner
December2016
Contents
1 ModelOrderReductionofIntegratedCircuits
inElectricalNetworks....................................................... 1
MichaelHinze,MartinKunkel,UlrichMatthes,
andMortenVierling
1.1 Introduction ............................................................. 1
1.2 BasicModels............................................................ 3
1.2.1 Coupling ........................................................ 4
1.3 SimulationoftheFullSystem.......................................... 7
1.3.1 StandardGalerkinFiniteElementApproach.................. 7
1.3.2 MixedFiniteElementApproach............................... 8
1.4 ModelOrderReductionUsingPOD................................... 13
1.4.1 NumericalInvestigation........................................ 16
1.4.2 NumericalInvestigation,PositionoftheSemiconductor
intheNetwork .................................................. 17
1.4.3 MORfortheNonlinearitywithDEIM ........................ 19
1.4.4 NumericalImplementationandResultswithDEIM.......... 20
1.5 Residual-BasedSampling .............................................. 23
1.5.1 NumericalInvestigationforResidualBasedSampling....... 26
1.6 PABTECCombinedwithPODMOR.................................. 27
1.6.1 Decoupling...................................................... 28
1.6.2 ModelReductionApproach.................................... 30
1.6.3 NumericalExperiments ........................................ 31
References..................................................................... 34
2 Element-BasedModelReductioninCircuitSimulation................. 39
AndreasSteinbrecherandTatjanaStykel
2.1 Introduction ............................................................. 39
2.2 CircuitEquations........................................................ 40
2.2.1 Graph-TheoreticConcepts ..................................... 41
2.2.2 ModifiedNodalAnalysisandModifiedLoopAnalysis ...... 41
2.2.3 LinearRLCCircuits............................................ 45
ix
x Contents
2.3 ModelReductionofLinearCircuits ................................... 47
2.3.1 BalancedTruncationforRLCCircuits ........................ 48
2.3.2 BalancedTruncationforRCCircuits.......................... 52
2.3.3 NumericalAspects.............................................. 59
2.4 ModelReductionofNonlinearCircuits................................ 61
2.5 SolvingMatrixEquations .............................................. 66
2.5.1 ADIMethodforProjectedLyapunovEquations.............. 67
2.5.2 Newton’sMethodforProjectedRiccatiEquations............ 68
2.6 MATLABToolboxPABTEC........................................... 71
2.7 NumericalExamples.................................................... 75
References..................................................................... 82
3 ReducedRepresentationofPowerGridModels.......................... 87
PeterBennerandAndréSchneider
3.1 Introduction ............................................................. 87
3.2 SystemDescription ..................................................... 89
3.2.1 BasicDefinitions................................................ 89
3.2.2 BenchmarkSystems............................................ 94
3.3 TerminalReductionApproaches....................................... 96
3.3.1 (E)SVDMOR ................................................... 96
3.3.2 TermMerg....................................................... 101
3.3.3 SparseRC........................................................ 103
3.3.4 MORforManyTerminalsviaInterpolation................... 106
3.4 ESVDMORinDetail ................................................... 108
3.4.1 Stability,Passivity,Reciprocity ................................ 108
3.4.2 ErrorAnalysis................................................... 114
3.4.3 ImplementationDetails......................................... 120
3.5 SummaryandOutlook.................................................. 131
References..................................................................... 132
4 Coupling of Numeric/Symbolic Reduction Methods
forGeneratingParametrizedModelsofNanoelectronic
Systems........................................................................ 135
OliverSchmidt,MatthiasHauser,andPatrickLang
4.1 Introduction ............................................................. 135
4.1.1 SymbolicModelingofAnalogCircuits ....................... 137
4.2 HierarchicalModellingandModelReduction......................... 137
4.2.1 WorkflowforSubsystemReductions.......................... 138
4.2.2 SubsystemSensitivities......................................... 140
4.2.3 SubsystemRanking............................................. 142
4.2.4 AlgorithmforHierarchicalModelReduction................. 144
4.3 Implementations......................................................... 144
4.4 Applications............................................................. 146
4.4.1 DifferentialAmplifier .......................................... 146
4.4.2 ReductionoftheTransmissionLineL1byUsingan
AdaptedPABTECAlgorithm.................................. 149
